let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = n |^ 5 ) implies for n being Nat holds (Partial_Sums s) . n = (((n |^ 2) * ((n + 1) |^ 2)) * (((2 * (n |^ 2)) + (2 * n)) - 1)) / 12 )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = ((($1 |^ 2) * (($1 + 1) |^ 2)) * (((2 * ($1 |^ 2)) + (2 * $1)) - 1)) / 12;
assume A1: for n being Nat holds s . n = n |^ 5 ; :: thesis: for n being Nat holds (Partial_Sums s) . n = (((n |^ 2) * ((n + 1) |^ 2)) * (((2 * (n |^ 2)) + (2 * n)) - 1)) / 12
A2: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume (Partial_Sums s) . n = (((n |^ 2) * ((n + 1) |^ 2)) * (((2 * (n |^ 2)) + (2 * n)) - 1)) / 12 ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = ((((n |^ 2) * ((n + 1) |^ 2)) * (((2 * (n |^ 2)) + (2 * n)) - 1)) / 12) + (s . (n + 1)) by SERIES_1:def 1
.= ((((n |^ 2) * ((n + 1) |^ 2)) * (((2 * (n |^ 2)) + (2 * n)) - 1)) / 12) + ((n + 1) |^ 5) by A1
.= ((((n |^ 2) * ((n + 1) |^ 2)) * (((2 * (n |^ 2)) + (2 * n)) - 1)) + (((n + 1) |^ (3 + 2)) * 12)) / 12
.= ((((n |^ 2) * ((n + 1) |^ 2)) * (((2 * (n |^ 2)) + (2 * n)) - 1)) + ((((n + 1) |^ 3) * ((n + 1) |^ 2)) * 12)) / 12 by NEWTON:8
.= (((n + 1) |^ 2) * (((n |^ 2) * (((2 * (n |^ 2)) + (2 * n)) - 1)) + (((n + 1) |^ 3) * 12))) / 12
.= (((n + 1) |^ 2) * (((n + 2) |^ 2) * (((2 * ((n + 1) |^ 2)) + (2 * (n + 1))) - 1))) / 12 by Lm15
.= ((((n + 1) |^ 2) * ((n + 2) |^ 2)) * (((2 * ((n + 1) |^ 2)) + (2 * (n + 1))) - 1)) / 12 ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= 0 |^ 5 by A1
.= ((0 * ((0 + 1) |^ 2)) * (((2 * (0 |^ 2)) + (2 * 0)) - 1)) / 12 by NEWTON:11
.= (((0 |^ 2) * ((0 + 1) |^ 2)) * (((2 * (0 |^ 2)) + (2 * 0)) - 1)) / 12 by NEWTON:11 ;
then A3: S₁[ 0 ] ;
for n being Nat holds S₁[n] from NAT_1:sch 2(A3, A2);
hence for n being Nat holds (Partial_Sums s) . n = (((n |^ 2) * ((n + 1) |^ 2)) * (((2 * (n |^ 2)) + (2 * n)) - 1)) / 12 ; :: thesis: verum